5.2 Far-shell integrals
Let \(E_r(n,t):=q_{\mathrm{sm}}^{4t}+q_{\mathrm{big}}^{4t}+e^{-a_r t n^{-2/3}}\). Then for every \(\delta \in (0,\pi /4)\),
\[ \begin{aligned} K_n\int _{\mathcal{B}_\delta \setminus \mathcal{D}_r}|\psi (\lambda )|^{4t}\, d\lambda & \le 2^{-n+1}E_r(n,t),\\ (2\pi )^{-d}\sum _{\lambda \in \Lambda } \int _{\lambda +(\mathcal{B}_{\pi /4}\setminus \mathcal{D}_r)}|\psi (\gamma )|^{4t}\, d\gamma & \le 2^{-n+1}E_r(n,t). \end{aligned} \]
Both are \(o(\hat A_{n,4t})\) as \(n\to \infty \) with \(t/(n^{8/3}\log t)\to \infty \).
Proof
Integrate the pointwise bounds from Proposition 23 over \(\mathcal{B}_{\pi /4}\setminus \mathcal{D}_r\) (volume \(\le (\pi /2)^d\)). For the translated shell, the multiplicativity \(\psi (\lambda +\mu )=\psi (\lambda )\psi (\mu )\) and \(|\psi (\lambda )|=1\) for \(\lambda \in \Lambda \) reduce to the same integral. The ratio \(2^{-n+1}E_r/\hat A_{n,4t}\to 0\) since \(t/(n^{8/3}\log t)\to \infty \) ensures \(tn^{-2/3}\gg d\log t\).